Dynamics on the attractor for the complex Ginzburg-Landau equation [report]

P. Takac
1994 unpublished
We present a numerical study of the large-time asymptotic behavior of solutions to the one-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. Our parameters belong to the Benjamin-Felt unstable region. Our solutions start near a pure-mode rotating wave that is stable under sideband perturbations for the Reynolds number R ranging over an interval (Rsub, Rsup). We find sub-and super-critical bifurcations from this stable rotating wave to a stable 2-torus as the
more » ... torus as the parameter a_ is decreased or increased past the critical value .Rsub or a_3u p. As R > Rsup further increases, we observe a variety of dynamical phenomena, such as a local attractor consisting of three unstable manifolds of periodic orbits or 2-tori cyclically connected by manifolds of connection orbits. We compare our numerical simulations to both rigorous mathematical results and experimental observations for binary fluid mixtures. Running head: Dynamics of the Ginzburg-Landau equation AMS(MOS) subject classifications: 76E30, 58F12 DISCLAIMER This reportwaspreparedas an accountof worksponsored by an agencyof the UnitedStates UI] "_'T_)J_ Gov©rnm¢nt. Neitherthe United States Government norany ag¢ncythereof,norany of their employees, makesany warranty,expressor implied,or assumesany legal liabilityor responsibility forthe accuracy, completeness, or usefulnessof any information, apparatus, product, or processdisclosed,or representsthat its use wouldnot infringeprivatelyownedrights.Refer-The submitted manuscript hasbeen authored ence hereinto any specificcommercialproduct, process, or serviceby tradename,trademark, by a contractor of the U.S. Government manufacturer, or otherwisedoes not necessarilyconstituteor imply its endorsement, recom-under comr_t No. W.31-109-_:NG-38. Accordin@_v. the U. S. Government retains a mendation,or favoringby the United States Governmentor any agency thereof.The views nonexclusive, royalty-free licvnse to _blish and opinionsof amhors expressedherein do not necessarilystate or reflect those of the or r_roducethe published form of this _t contribution, or allow others to do so, for UnitodState, s Government or an),agency thcroof. _ U.S. Government purposes. oii_I_JlIIJIJTIOH OF rills DOCUMENT18 LINLIMITEII t i
doi:10.2172/10174640 fatcat:rzxqmebipnhspj6ahwlg3rs7wa